Imperfect Price Discrimination profit per group

Question

During imperfect price discrimination is it possible to determine profit per group. For example:

Aggregated Cost function: C(Q) = 18*Q + 5; where Q = q1 + q2

Aggregated Demand (AD) for group 1: q1(p1) = 8 - [(1/6)*p1]

AD for group 2: q2(P2) = 4 - [(1/6)*p2]

After finding the inverse demand function (p1 and p2), setting up the profit maximization, and solving for q*1 and q*2, p*1 and p*2, and profit max., can one compute the individual group profit for group 1 and 2?

In order to solve this condition one would need the aggregated cost for each group. Because the monopoly is creating and selling the same good the cost would be the same between the two groups. Is it possible to divide the aggregated market cost function by the number of groups (in this case by two) to find the group's aggregated cost function?

In the end, the sum of the profits from group 1 and 2 should equal the profit maximization solved earlier.

Any direction (i.g. books or research papers) would be great as well! Thank you in advance.

Answer 1

No, you cannot clearly determine the individual cost functions. For example both the individual cost functions $$ C_1(q_1) = 18q_1 + 3, \hskip 20pt C_2(q_2) = 18q_2 + 2 $$ and $$ \hat{C}_1(q_1) = 18q_1 + 1, \hskip 20pt \hat{C}_2(q_2) = 18q_2 + 4 $$ would yield the same optimum and sum up to $C(Q) = 18Q + 5$.

Fixed costs can be allocated to any items, they do not change production decisions. Even in accounting there are several methods for actually allocating common fixed costs between production units.